Question

If a,b,c are in AP as well as in GP, then find the value of a^b-c + b^c-a + c^a-b

Answer 1

a, b , c are in AP as well as GP ,
it means a , b , c are follow the both conditions ( AP and GP) .
e.g b = (a + c)/2 { from AP condition}------(1)
b² = ac { from GP condition }-------(2)

now, put eqn (1) into eqn (2)

(a + c)²/4 = ac
(a + c)² = 4ac
a² +c² + 2ac = 4ac
a² + c² - 2ac = 0
(a - c)² = 0
a - c = 0 => a = c
hence, b = (a + c)/2 = a = c
hence , a = b = c ---------------(3)

now,
a^(b - c) + b^(c - a) + c^(a - b)
from eqn (3)
a^(b - b) + b^(c - c) + c^(a - a)
= a^0 + b^0 + c^0
= 1 + 1 + 1 = 3

Answer 2

As a,b,c are both in AP as well as GP

It follows some CONDITIONS which are:

b=(a+c)/2(AP CONDITION)--------------(1)

b²=ac(GP CONDITION)-------------(2)

PUTTING (1) IN (2)

(a+c)²/4=ac

=>(a+c)²=4ac

=>a²+2ac+c²=4ac

=>a²+c²=4ac-2ac

=>a²+c²=2ac

=>a²-2ac+c²=0

=>(a-c)²=0

=>a-c=0

=>a=c

From(1)

b=(a+c)/2

=>b=2c/2

=>b=c

.

. . b=c=a-----(3)

From(3)

a^b-b+b^c-c+c^a-a

=>a⁰+b⁰+c⁰

=>1+1+1

=>3(answer)

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